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Reducing Rational Expressions to Lowest Terms

Each rational number can be written in infinitely many equivalent forms. For
example,

Each equivalent form of
is obtained from
by multiplying both numerator and
denominator by the same nonzero number. For example,

Note that we are actually multiplying
by equivalent forms of 1, the multiplicative
identity.
If we start with
and convert it into
, we are simplifying by reducing
to its
lowest terms.We can reduce as follows:

A rational number is expressed in its lowest terms when the numerator and denominator
have no common factors other than 1. In reducing
, we divide the numerator
and denominator by the common factor 2, or â€œdivide outâ€ the common factor 2. We
can multiply or divide both numerator and denominator of a rational number by the
same nonzero number without changing the value of the rational number. This fact
is called the basic principle of rational numbers.

Basic Principle of Rational Numbers

If
is a rational number and c is a nonzero real number, then

Helpful hint

Most students learn to convert
into
by dividing 3 into
6 to get 2 and then multiply 2
by 2 to get 4. In algebra it is
better to do this conversion
by multiplying the numerator
and denominator of
by 2 as shown here.

Caution

Although it is true that

we cannot divide out the 2â€™s in this expression because the 2â€™s are not factors. We
can divide out only common factors when reducing fractions.

Just as a rational number has infinitely many equivalent forms, a rational expression
also has infinitely many equivalent forms. To reduce rational expressions
to its lowest terms, we follow exactly the same procedure as we do for rational
numbers: Factor the numerator and denominator completely, then divide out all
common factors.

Example 1

Reducing

Reduce each rational expression to its lowest terms.

Solution

a) Factor 18 as 2 Â· 32 and 42 as 2 Â· 3 Â· 7:

Factor.

Divide out the common factors.

b) Because this expression is already factored, we use the quotient rule for exponents
to reduce: